Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition II. Portfolio Theory 5. Risk and Return: Past and Prologue © The McGraw−Hill Companies, 2003 the slope of the CML based on the subperiod data. Indeed, the differences across subperiods are quite striking. The most plausible explanation for the variation in subperiod returns is based on the observation that the standard deviation of returns is quite large in all subperiods. If we take the 76-year standard deviation of 20.3% as representative and assume that returns in one year are nearly uncorrelated with those in other years (the evidence suggests that any correlation across years is small), then the standard deviation of our estimate of the mean return in any of our 19-year subperiods will be 20.3/ , which is fairly large. This means that in approximately one out of three cases, a 19-year average will deviate by 4.7% or more from the true mean. Applying this insight to the data in Table 5.5 tells us that we cannot reject with any confidence the possibility that the true mean is similar in all subperiods! In other words, the “noise” in the data is so large that we simply cannot make reliable inferences from average re- turns in any subperiod. The variation in returns across subperiods may simply reflect statisti- cal variation, and we have to reconcile ourselves to the fact that the market return and the reward-to-variability ratio for passive (as well as active!) strategies is simply very hard to predict. The instability of average excess return on stocks over the 19-year subperiods in Table 5.5 also calls into question the precision of the 76-year average excess return (8.64%) as an esti- mate of the risk premium on stocks looking into the future. In fact, there has been consider- able recent debate among financial economists about the “true” equity risk premium, with an emerging consensus that the historical average is an unrealistically high estimate of the future risk premium. This argument is based on several factors: the use of longer time periods in ͙ළළ19 ϭ 4.7% 157 Triumph of the Optimists As a whole, the last 7 decades have been very kind to U.S. equity investors. Stock investments have out- performed investments in safe Treasury bills by more than 8% per year. The real rate of return averaged more than 9%, implying an expected doubling of the real value of the investment portfolio about every 8 years! Is this experience representative? A new book by three professors at the London Business School, Elroy Dimson, Paul Marsh, and Mike Staunton, extends the U.S. evidence to other countries and to longer time periods. Their conclusion is given in the book’s title, Triumph of the Optimists*: in every country in their study (which included markets in North America, Eu- rope, Asia, and Africa), the investment optimists—those who bet on the economy by investing in stocks rather than bonds or bills—were vindicated. Over the long haul, stocks beat bonds everywhere. On the other hand, the equity risk premium is prob- ably not as large as the post-1926 evidence from Table 5.1 would seem to indicate. First, results from the first 25 years of the last century (which included the first World War) were less favorable to stocks. Second, U.S. returns have been better than that of most other countries, and so a more representative value for the historical risk premium may be lower than the U.S. ex- perience. Finally, the sample that is amenable to his- torical analysis suffers from a self-selection problem. Only those markets that have survived to be studied can be included in the analysis. This leaves out coun- tries such as Russia or China, whose markets were shut down during communist rule, and whose results if included would surely bring down the average perfor- mance of equity investments. Nevertheless, there is powerful evidence of a risk premium that shows its force everywhere the authors looked. *Elroy Dimson, Paul Marsh, Mike Staunton, Triumph of the Optimists: 101 Years of Global Investment Returns. Princeton University Press, Princeton, N.J.: 2002. Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition II. Portfolio Theory 5. Risk and Return: Past and Prologue © The McGraw−Hill Companies, 2003 which equity returns are examined; a broad range of countries rather than just the U.S. in which excess returns are computed (Dimson, Marsh, and Staunton, 2001); direct surveys of financial executives about their expectations for stock market returns (Graham and Harvey, 2001); and inferences from stock market data about investor expectations (Jagannathan, McGrattan, and Scherbina, 2000; Fama and French, 2002). The nearby box discusses some of this evidence. Costs and Benefits of Passive Investing How reasonable is it for an investor to pursue a passive strategy? We cannot answer such a question definitively without comparing passive strategy results to the costs and benefits ac- cruing to an active portfolio strategy. Some issues are worth considering, however. First, the alternative active strategy entails costs. Whether you choose to invest your own valuable time to acquire the information needed to generate an optimal active portfolio of risky assets or whether you delegate the task to a professional who will charge a fee, con- structing an active portfolio is more expensive than constructing a passive one. The passive portfolio requires only small commissions on purchases of U.S. T-bills (or zero commissions if you purchase bills directly from the government) and management fees to a mutual fund company that offers a market index fund to the public. An index fund has the lowest operating expenses of all mutual stock funds because it requires minimal effort. A second argument supporting a passive strategy is the free-rider benefit. If you assume there are many active, knowledgeable investors who quickly bid up prices of undervalued as- sets and offer down overvalued assets (by selling), you have to conclude that most of the time most assets will be fairly priced. Therefore, a well-diversified portfolio of common stock will be a reasonably fair buy, and the passive strategy may not be inferior to that of the average ac- tive investor. We will expand on this insight and provide a more comprehensive analysis of the relative success of passive strategies in Chapter 8. To summarize, a passive strategy involves investment in two passive portfolios: virtually risk-free short-term T-bills (or a money market fund) and a fund of common stocks that mim- ics a broad market index. Recall that the capital allocation line representing such a strategy is called the capital market line. Using Table 5.5, we see that using 1926 to 2001 data, the pas- sive risky portfolio has offered an average excess return of 8.6% with a standard deviation of 20.7%, resulting in a reward-to-variability ratio of 0.42. 158 Part TWO Portfolio Theory SUMMARY • Investors face a trade-off between risk and expected return. Historical data confirm our intuition that assets with low degrees of risk provide lower returns on average than do those of higher risk. • Shifting funds from the risky portfolio to the risk-free asset is the simplest way to reduce risk. Another method involves diversification of the risky portfolio. We take up diversification in later chapters. • U.S. T-bills provide a perfectly risk-free asset in nominal terms only. Nevertheless, the standard deviation of real rates on short-term T-bills is small compared to that of assets such as long-term bonds and common stocks, so for the purpose of our analysis, we consider T-bills the risk-free asset. Besides T-bills, money market funds hold short-term, safe obligations such as commercial paper and CDs. These entail some default risk but relatively little compared to most other risky assets. For convenience, we often refer to money market funds as risk-free assets. • A risky investment portfolio (referred to here as the risky asset) can be characterized by its reward-to-variability ratio. This ratio is the slope of the capital allocation line (CAL), the www.mhhe.com/bkm Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition II. Portfolio Theory 5. Risk and Return: Past and Prologue © The McGraw−Hill Companies, 2003 5 Risk and Return: Past and Prologue 159 www.mhhe.com/bkm line connecting the risk-free asset to the risky asset. All combinations of the risky and risk- free asset lie on this line. Investors would prefer a steeper sloping CAL, because that means higher expected returns for any level of risk. If the borrowing rate is greater than the lending rate, the CAL will be “kinked” at the point corresponding to an investment of 100% of the complete portfolio in the risky asset. • An investor’s preferred choice among the portfolios on the capital allocation line will depend on risk aversion. Risk-averse investors will weight their complete portfolios more heavily toward Treasury bills. Risk-tolerant investors will hold higher proportions of their complete portfolios in the risky asset. • The capital market line is the capital allocation line that results from using a passive investment strategy that treats a market index portfolio, such as the Standard & Poor’s 500, as the risky asset. Passive strategies are low-cost ways of obtaining well-diversified portfolios with performance that will reflect that of the broad stock market. KEY TERMS arithmetic average, 133 asset allocation, 148 capital allocation line, 152 capital market line, 156 complete portfolio, 149 dollar-weighted average return, 134 excess return, 138 expected return, 136 geometric average, 133 holding-period return, 132 inflation rate, 147 nominal interest rate, 147 passive strategy, 156 probability distribution, 136 real interest rate, 147 reward-to-variability ratio, 152 risk aversion, 138 risk-free rate, 137 risk premium, 137 scenario analysis, 136 standard deviation, 136 variance, 136 PROBLEM SETS 1. A portfolio of nondividend-paying stocks earned a geometric mean return of 5.0% between January 1, 1996, and December 31, 2002. The arithmetic mean return for the same period was 6.0 %. If the market value of the portfolio at the beginning of 1996 was $100,000, what was the market value of the portfolio at the end of 2002? 2. Which of the following statements about the standard deviation is/are true? A standard deviation: i. Is the square root of the variance. ii. Is denominated in the same units as the original data. iii. Can be a positive or a negative number. 3. Which of the following statements reflects the importance of the asset allocation decision to the investment process? The asset allocation decision: a. Helps the investor decide on realistic investment goals. b. Identifies the specific securities to include in a portfolio. c. Determines most of the portfolio’s returns and volatility over time. d. Creates a standard by which to establish an appropriate investment time horizon. 4. Look at Table 5.2 in the text. Suppose you now revise your expectations regarding the stock market as follows: State of the Economy Probability HPR Boom 0.3 44% Normal growth 0.4 14 Recession 0.3 Ϫ16 Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition II. Portfolio Theory 5. Risk and Return: Past and Prologue © The McGraw−Hill Companies, 2003 Use Equations 5.3–5.5 to compute the mean and standard deviation of the HPR on stocks. Compare your revised parameters with the ones in the text. 5. The stock of Business Adventures sells for $40 a share. Its likely dividend payout and end-of-year price depend on the state of the economy by the end of the year as follows: Dividend Stock Price Boom $2.00 $50 Normal economy 1.00 43 Recession .50 34 a. Calculate the expected holding-period return and standard deviation of the holding- period return. All three scenarios are equally likely. b. Calculate the expected return and standard deviation of a portfolio invested half in Business Adventures and half in Treasury bills. The return on bills is 4%. Use the following data in answering questions 6, 7, and 8. Utility Formula Data Expected Standard Investment Return E(r) Deviation ␴ 1 .12 .30 2 .15 .50 3 .21 .16 4 .24 .21 U ϭ E(r) Ϫ 1 ⁄2A␴ 2 where A ϭ 4 6. Based on the utility formula above, which investment would you select if you were risk averse with A ϭ 4? a. 1 b. 2 c. 3 d. 4 7. Based on the utility formula above, which investment would you select if you were risk neutral? a. 1 b. 2 c. 3 d. 4 8. The variable (A) in the utility formula represents the: a. investor’s return requirement. b. investor’s aversion to risk. c. certainty equivalent rate of the portfolio. d. preference for one unit of return per four units of risk. Use the following expectations on Stocks X and Y to answer questions 9 through 12 (round to the nearest percent). 160 Part TWO Portfolio Theory www.mhhe.com/bkm Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition II. Portfolio Theory 5. Risk and Return: Past and Prologue © The McGraw−Hill Companies, 2003 Bear Market Normal Market Bull Market Probability 0.2 0.5 0.3 Stock X Ϫ20% 18% 50% Stock Y Ϫ15% 20% 10% 9. What are the expected returns for Stocks X and Y? Stock X Stock Y a. 18% 5% b. 18% 12% c. 20% 11% d. 20% 10% 10. What are the standard deviations of returns on Stocks X and Y? Stock X Stock Y a. 15% 26% b. 20% 4% c. 24% 13% d. 28% 8% 11. Assume that of your $10,000 portfolio, you invest $9,000 in Stock X and $1,000 in Stock Y. What is the expected return on your portfolio? a. 18% b. 19% c. 20% d. 23% 12. Probabilities for three states of the economy, and probabilities for the returns on a particular stock in each state are shown in the table below. Probability of Stock Performance Probability of Stock in Given State of Economy Economic State Performance Economic State Good .3 Good .6 Neutral .3 Poor .1 Neutral .5 Good .4 Neutral .3 Poor .3 Poor .2 Good .2 Neutral .3 Poor .5 5 Risk and Return: Past and Prologue 161 www.mhhe.com/bkm Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition II. Portfolio Theory 5. Risk and Return: Past and Prologue © The McGraw−Hill Companies, 2003 The probability that the economy will be neutral and the stock will experience poor performance is a. .06 c. .50 b. .15 d. .80 13. An analyst estimates that a stock has the following probabilities of return depending on the state of the economy: State of Economy Probability Return Good .1 15% Normal .6 13 Poor .3 7 The expected return of the stock is: a. 7.8% b. 11.4% c. 11.7% d. 13.0% 14. XYZ stock price and dividend history are as follows: Year Beginning-of-Year Price Dividend Paid at Year-End 1999 $100 $4 2000 $110 $4 2001 $ 90 $4 2002 $ 95 $4 An investor buys three shares of XYZ at the beginning of 1999 buys another two shares at the beginning of 2000, sells one share at the beginning of 2001, and sells all four remaining shares at the beginning of 2002. a. What are the arithmetic and geometric average time-weighted rates of return for the investor? b. What is the dollar-weighted rate of return. Hint: Carefully prepare a chart of cash flows for the four dates corresponding to the turns of the year for January 1, 1999, to January 1, 2002. If your calculator cannot calculate internal rate of return, you will have to use trial and error. 15. a. Suppose you forecast that the standard deviation of the market return will be 20% in the coming year. If the measure of risk aversion in equation 5.6 is A ϭ 4, what would be a reasonable guess for the expected market risk premium? b. What value of A is consistent with a risk premium of 9%? c. What will happen to the risk premium if investors become more risk tolerant? 16. Using the historical risk premiums as your guide, what is your estimate of the expected annual HPR on the S&P 500 stock portfolio if the current risk-free interest rate is 5%? 17. What has been the historical average real rate of return on stocks, Treasury bonds, and Treasury notes? 18. Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $50,000 or $150,000, with equal probabilities of 0.5. The alternative riskless investment in T-bills pays 5%. 162 Part TWO Portfolio Theory www.mhhe.com/bkm Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition II. Portfolio Theory 5. Risk and Return: Past and Prologue © The McGraw−Hill Companies, 2003 a. If you require a risk premium of 10%, how much will you be willing to pay for the portfolio? b. Suppose the portfolio can be purchased for the amount you found in (a). What will the expected rate of return on the portfolio be? c. Now suppose you require a risk premium of 15%. What is the price you will be willing to pay now? d. Comparing your answers to (a) and (c), what do you conclude about the relationship between the required risk premium on a portfolio and the price at which the portfolio will sell? For problems 19–23, assume that you manage a risky portfolio with an expected rate of re- turn of 17% and a standard deviation of 27%. The T-bill rate is 7%. 19. a. Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund. What is the expected return and standard deviation of your client’s portfolio? b. Suppose your risky portfolio includes the following investments in the given proportions: Stock A 27% Stock B 33% Stock C 40% What are the investment proportions of your client’s overall portfolio, including the position in T-bills? c. What is the reward-to-variability ratio (S) of your risky portfolio and your client’s overall portfolio? d. Draw the CAL of your portfolio on an expected return/standard deviation diagram. What is the slope of the CAL? Show the position of your client on your fund’s CAL. 20. Suppose the same client in problem 19 decides to invest in your risky portfolio a proportion (y) of his total investment budget so that his overall portfolio will have an expected rate of return of 15%. a. What is the proportion y? b. What are your client’s investment proportions in your three stocks and the T-bill fund? c. What is the standard deviation of the rate of return on your client’s portfolio? 21. Suppose the same client in problem 19 prefers to invest in your portfolio a proportion (y) that maximizes the expected return on the overall portfolio subject to the constraint that the overall portfolio’s standard deviation will not exceed 20%. a. What is the investment proportion, y? b. What is the expected rate of return on the overall portfolio? 22. You estimate that a passive portfolio invested to mimic the S&P 500 stock index yields an expected rate of return of 13% with a standard deviation of 25%. Draw the CML and your fund’s CAL on an expected return/standard deviation diagram. a. What is the slope of the CML? b. Characterize in one short paragraph the advantage of your fund over the passive fund. 23. Your client (see problem 19) wonders whether to switch the 70% that is invested in your fund to the passive portfolio. a. Explain to your client the disadvantage of the switch. 5 Risk and Return: Past and Prologue 163 www.mhhe.com/bkm Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition II. Portfolio Theory 5. Risk and Return: Past and Prologue © The McGraw−Hill Companies, 2003 b. Show your client the maximum fee you could charge (as a percent of the investment in your fund deducted at the end of the year) that would still leave him at least as well off investing in your fund as in the passive one. (Hint: The fee will lower the slope of your client’s CAL by reducing the expected return net of the fee.) 24. What do you think would happen to the expected return on stocks if investors perceived an increase in the volatility of stocks? 25. The change from a straight to a kinked capital allocation line is a result of the: a. Reward-to-variability ratio increasing. b. Borrowing rate exceeding the lending rate. c. Investor’s risk tolerance decreasing. d. Increase in the portfolio proportion of the risk-free asset. 26. You manage an equity fund with an expected risk premium of 10% and an expected standard deviation of 14%. The rate on Treasury bills is 6%. Your client chooses to invest $60,000 of her portfolio in your equity fund and $40,000 in a T-bill money market fund. What is the expected return and standard deviation of return on your client’s portfolio? Expected Return Standard Deviation of Return a. 8.4% 8.4% b. 8.4 14.0 c. 12.0 8.4 d. 12.0 14.0 27. What is the reward-to-variability ratio for the equity fund in problem 26? a. .71 b. 1.00 c. 1.19 d. 1.91 For problems 28–30, download Table 5.3: Rates of return, 1926–2001, from www.mhhe.com/ blkm. 28. Calculate the same subperiod means and standard deviations for small stocks as Table 5.5 of the text provides for large stocks. a. Do small stocks provide better reward-to-variability ratios than large stocks? b. Do small stocks show a similar declining trend in standard deviation as Table 5.5 documents for large stocks? 29. Convert the nominal returns on both large and small stocks to real rates. Reproduce Table 5.5 using real rates instead of excess returns. Compare the results to those of Table 5.5. 30. Repeat problem 29 for small stocks and compare with the results for nominal rates. 164 Part TWO Portfolio Theory www.mhhe.com/bkm Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition II. Portfolio Theory 5. Risk and Return: Past and Prologue © The McGraw−Hill Companies, 2003 5 Risk and Return: Past and Prologue 165 www.mhhe.com/bkm SOLUTIONS TO 1. a. The arithmetic average is (2 ϩ 8 Ϫ 4)/3 ϭ 2% per month. b. The time-weighted (geometric) average is [(1 ϩ .02) ϫ (1 ϩ .08) ϫ (1 Ϫ .04)] 1/3 ϭ .0188 ϭ 1.88% per month c. We compute the dollar-weighted average (IRR) from the cash flow sequence (in $ millions): Month 12 3 Assets under management at beginning of month 10.0 13.2 19.256 Investment profits during month (HPR ϫ Assets) 0.2 1.056 (0.77) Net inflows during month 3.0 5.0 0.0 Assets under management at end of month 13.2 19.256 18.486 Time 01 2 3 Net cash flow * Ϫ10 Ϫ3.0 Ϫ5.0 ϩ18.486 * Time 0 is today. Time 1 is the end of the first month. Time 3 is the end of the third month, when net cash flow equals the ending value (potential liquidation value) of the portfolio. The IRR of the sequence of net cash flows is 1.17% per month. The dollar-weighted average is less than the time-weighted average because the negative return was realized when the fund had the most money under management. Concept CHECKS < WEBMASTER Inflation and Interest Rates The Federal Reserve Bank of St. Louis has several sources of information available on interest rates and economic conditions. One publication called Monetary Trends contains graphs and tabular information relevant to assess conditions in the capital markets. Go to the most recent edition of Monetary Trends at http://www .stls.frb.org/ docs/publications/mt/mt.pdf and answer the following questions: 1. What is the current level of three-month and long-term Treasury yields? 2. Have nominal interest rates increased, decreased, or remained the same over the last three months? 3. Have real interest rates increased, decreased, or remained the same over the last two years? 4. Examine the information comparing recent U.S. inflation and long-term interest rates with the inflation and long-term interest rate experience of Japan. Are the results consistent with theory? Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition II. Portfolio Theory 5. Risk and Return: Past and Prologue © The McGraw−Hill Companies, 2003 2. Computing the HPR for each scenario we convert the price and dividend data to rate of return data: Business Conditions Probability HPR High growth 0.35 67.66% ϭ (4.40 ϩ 35 Ϫ 23.50)/23.50 Normal growth 0.30 31.91% ϭ (4.00 ϩ 27 Ϫ 23.50)/23.50 No growth 0.35 Ϫ19.15% ϭ (4.00 ϩ 15 Ϫ 23.50)/23.50 Using Equations 5.1 and 5.2 we obtain E(r) ϭ 0.35 ϫ 67.66 ϩ 0.30 ϫ 31.91 ϩ 0.35 ϫ (Ϫ19.15) ϭ 26.55% ␴ 2 ϭ 0.35 ϫ (67.66 Ϫ 26.55) 2 ϩ 0.30 ϫ (31.91 Ϫ 26.55) 2 ϩ 0.35 ϫ (Ϫ19.15 Ϫ 26.55) 2 ϭ 1331 and ␴ϭ͙ළළළළ1331 ϭ 36.5% 3. If the average investor chooses the S&P 500 portfolio, then the implied degree of risk aversion is given by Equation 5.7: A ϭϭ3.09 4. The mean excess return for the period 1926–1934 is 3.56% (below the historical average), and the standard deviation (using n Ϫ 1 degrees of freedom) is 32.69% (above the historical average). These results reflect the severe downturn of the great crash and the unusually high volatility of stock returns in this period. 5. a. Solving 1 ϩ R ϭ (1 ϩ r)(1 ϩ i) ϭ (1.03)(1.08) ϭ 1.1124 R ϭ 11.24% b. Solving 1 ϩ R ϭ (1.03)(1.10) ϭ 1.133 R ϭ 13.3% 6. Holding 50% of your invested capital in Ready Assets means your investment proportion in the risky portfolio is reduced from 70% to 50%. Your risky portfolio is constructed to invest 54% in Vanguard and 46% in Fidelity. Thus, the proportion of Vanguard in your overall portfolio is 0.5 ϫ 54% ϭ 27%, and the dollar value of your position in Vanguard is 300,000 ϫ 0.27 ϭ $81,000. 7. E(r) ϭ 7 ϩ 0.75 ϫ 8% ϭ 13% ␴ϭ0.75 ϫ 22% ϭ 16.5% Risk premium ϭ 13 Ϫ 7 ϭ 6% ϭϭ.36 13 Ϫ 7 16.5 Risk premium Standard deviation .10 Ϫ .05 1 ⁄2 ϫ .18 2 166 Part TWO Portfolio Theory www.mhhe.com/bkm [...]... must do, however, is to show investors the entire investment opportunity set as we do in Figure 6 .3 This is the set of all attainable combinations of risk and return offered by portfolios formed using the available assets in differing proportions Points on the investment opportunity set of Figure 6 .3 can be found by varying the investment proportions and computing the resulting expected returns and standard... risk-free asset First, we offer an overview As in the two-risky-assets example, the problem has three separate steps To begin, we identify the best possible or most efficient risk-return combinations available from the universe of risky assets Next we determine the optimal portfolio of risky assets by finding the portfolio that supports the steepest CAL Finally, we choose an appropriate complete portfolio... McGraw−Hill Companies, 20 03 The risk measure is based on the concept of value at risk and includes some capabilities of stress testing http://aida.econ.yale.edu/~shiller/data.htm Professor Shiller provides historical data used in his applications in Irrational Exuberance The site also has links to other data sites http://www.mhhe.com/edumarketinsight The Education Version of Market Insight contains information... market securities versus in a risky portfolio We simply assumed that the risky portfolio comprised a stock and a bond fund in given proportions Of course, investors need to decide on the proportion of their portfolios to allocate to the stock versus the bond market This, too, is an asset allocation decision As the box on page 1 73 emphasizes, most investment professionals recognize that the asset allocation... 10.81 83% 10.81 83% The Efficient Frontier of Risky Assets To get a sense of how additional risky assets can improve the investor s investment opportunities, look at Figure 6.9 Points A, B, and C represent the expected returns and standard deviations of three stocks The curve passing through A and B shows the risk-return combinations of all the portfolios that can be formed by combining those two stocks Similarly,... being diversified, because the idea of investing in a mix of stocks, bonds and other financial assets meant missing out on some of the soaring gains of tech stocks But with the collapse of the tech bubble and now the fall of Enron Corp wiping out the 401(k) holdings of many current and retired Enron employees, the dangers of overloading a portfolio with one stock—or even with a group of similar stocks—has... investment spread across many securities, exposure to any particular source of risk is negligible This is just an application of the law of averages The reduction of risk to very low levels because of independent risk sources is sometimes called the insurance principle When common sources of risk affect all firms, however, even extensive diversification cannot eliminate risk In Figure 6.1B, portfolio standard... Companies, 20 03 Dangers of Not Diversifying Hit Investors Enron, Tech Bubble Are Wake-Up Calls Mutual-fund firms and financial planners have droned on about the topic for years But suddenly, it s at the epicenter of lawsuits, congressional hearings and presidential reform proposals Diversification—that most basic of investing principles—has returned with a vengeance During the late 199 0s, many people scoffed... overwhelming tendency of the returns on the stock and bond funds to vary inversely in this scenario analysis We are now in a position to derive the risk and return features of portfolios of risky assets 1 Suppose the rates of return of the bond portfolio in the three scenarios of Spreadsheet 6.4 are 10% in a recession, 7% in a normal period, and ϩ2% in a boom The stock returns in the three scenarios are... average of the expected returns on the component securities, with the same portfolio proportions as weights In symbols, the expectation of Equation 6.1 is E(rP) ϭ wBE(rB) ϩ wSE(rS) (6.2) The first two rules are simple linear expressions This is not so in the case of the portfolio variance, as the third rule shows Rule 3: The variance of the rate of return on the two-risky-asset portfolio is 2 P ϭ (wB . ac- tive investor. We will expand on this insight and provide a more comprehensive analysis of the relative success of passive strategies in Chapter 8. To summarize, a passive strategy involves investment. concept of value at risk and includes some capabilities of stress testing. http://aida.econ.yale.edu/~shiller/data.htm Professor Shiller provides historical data used in his applications in Irrational. portfolio? b. Suppose your risky portfolio includes the following investments in the given proportions: Stock A 27% Stock B 33 % Stock C 40% What are the investment proportions of your client s overall portfolio,